Stochastic optimal control with control-dependent diffusion and state constraints: A degenerate elliptic approach

We study a stochastic optimal control problem where the state process is constrained to remain within a smooth, compact domain. The control influences both the drift and a possibly degenerate, control-dependent diffusion matrix, leading to a fully nonlinear, degenerate elliptic Hamilton–Jacobi–Bellman (HJB) equation with a nontrivial Neumann boundary condition. Although these aspects have been analyzed individually in previous works, this paper develops a unified framework that seeks to bring them together within a common formulation. We prove that the associated optimal value function is the unique viscosity solution of the HJB equation with the specified boundary condition. To illustrate the practical applicability of the approach, we provide an example that demonstrates the main features of the framework.

Recommended citation: Anderson O. Calixto, Bernardo Freitas Paulo da Costa, Glauco Valle. Stochastic optimal control with control-dependent diffusion and state constraints: A degenerate elliptic approach. Mathematical Control and Related Fields. https://doi.org/10.3934/mcrf.2026027
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